A zero of x³ + 64 is

If f and g are polynomials of degrees m and n respectively, and if h(x) = (f ^_o g)(x), then the degree of h is

The number of real numbers in [0, 2π] satisfying sin⁴x - 2sin²x + 1 is

If x³+12x²+10ax+1999 definitely has a positive zero, if and only if

The number of positive zeros of the polynomial \(\overset { n }{ \underset { j=0 }{ \Sigma } } { n }_{ C_{ r } }\)(-1)^rx^r is

If a, b, c ∈ Q and p +√q (p, q ∈ Q) is an irrational root of ax²+bx+c = 0 then the other root is ___________

Let a > 0, b > 0, c >0. Theh n both the root of the equation ax²+bx+c = 0 are _________

lf the root of the equation x³ + bx²+ cx - 1 = 0 form an lncreasing G.P, then ___________

For real x, the equation \(\left| \frac { x }{ x-1 } \right| +|x|=\frac { { x }^{ 2 } }{ |x-1| } \) has ________

If the equation ax²+ bx+c = 0(a > 0) has two roots ∝ and β such that ∝ <- 2 and β > 2, then __________

If \((2+\sqrt{3})^{x^{2}-2 x+1}+(2-\sqrt{3})^{x^{2}-2 x-1}=\frac{2}{2-\sqrt{3}}\) then x = _________

If ∝, β, ૪ are the roots of 9x³-7x+6 = 0, then ∝ β ૪ is __________

If x² - hx - 21 = 0 and x² - 3hx + 35 = 0 (h > 0) have a common root, then h = ___________

If ax² + bx + c = 0, a, b, c \(\in\) R has no real zeros, and if a + b + c < 0, then __________

If p(x) = ax² + bx + c and Q(x) = -ax² + dx + c where ac ≠ 0 then p(x). Q(x) = 0 has at least _______ real roots.

If α, β, γ  and \(\delta\) are the roots of the polynomial equation 2x⁴ + 5x³ − 7x² + 8 = 0, find a quadratic equation with integer coefficients whose roots are α + β + γ + \(\delta\) and αβ૪\(\delta\).

Show that the equation 2x²− 6x +7 = 0 cannot be satisfied by any real values of x.

If x²+2(k+2)x+9k = 0 has equal roots, find k.

Solve: (2x-1) (x+3) (x-2) (2x+3)+20 = 0

Determine the number of positive and negative roots of the equation x⁹- 5x⁸-14x⁷= 0.

Construct a cubic equation with roots 1, 1 and −2

Construct a cubic equation with roots 2, −2, and 4.

Discuss the nature of the roots of the following polynomials:
x⁵-19x⁴+ 2x³+ 5x²+11

Find the Interval for a for which 3x²+2(a²+1) x+(a²-3a+2) possesses roots of opposite sign.

Find x If \(x=\sqrt { 2+\sqrt { 2+\sqrt { 2+....+upto\infty } } } \)

If α and β are the roots of the quadratic equation 2x²−7x+13 = 0 , construct a quadratic equation whose roots are α² and β².

If the sides of a cubic box are increased by 1, 2, 3 units respectively to form a cuboid, then the volume is increased by 52 cubic units. Find the volume of the cuboid.

If α, β and γ are the roots of the cubic equation x³+2x²+3x+4 = 0, form a cubic equation whose roots are, 2α, 2β, 2γ

Find the sum of squares of roots of the equation 2x⁴- 8x³+ 6x²-3 = 0.

Find the condition that the roots of cubic x³+ ax²+ bx + c = 0 are in the ratio p : q : r.

If p is real, discuss the nature of the roots of the equation 4x²+ 4px + p + 2 = 0 in terms of p.

If α, β, and γ are the roots of the polynomial equation ax³+ bx²+ cx + d = 0, find the value of \(\Sigma \frac { \alpha }{ \beta \gamma } \) in terms of the coefficients.

Solve the equation
2x³ - 9x² + 10x = 3

Solve the following equations,
12x³+ 8x = 29x²- 4

Solve the equations
x⁴+ 3x³- 3x - 1 = 0

Form the equation whose roots are the squares of the roots of the cubic equation x³+ ax²+ bx + c = 0.

Solve the equation x³− 9x²+14x + 24 = 0 if it is given that two of its roots are in the ratio 3:2.

If 2+i and 3-\(\sqrt{2}\) are roots of the equation x⁶-13x⁵+ 62x⁴-126x³+ 65x²+127x-140 = 0, find all roots.

Solve : (x - 5) (x - 7) (x + 6) (x + 4) = 504

If a, b, c, d and p are distinct non-zero real numbers such that (a²+b²+c²) p²-2 (ab+bc+cd) p+(b²+c²+d²)≤ 0 then prove that a, b, c, d are in G.P and ad = bc

If c ≠ 0 and \(\frac { p }{ 2x } =\frac { a }{ x+x } +\frac { b }{ x-c } \) has two equal roots, then find p. 

Solve: (2x² - 3x + 1) (2x² + 5x + 1) = 9x².